Generalization of Menger’s result on the structure of logical formulas
Dal Charles Gerneth · Bulletin of the American Mathematical Society · 1948
Mengers paper 2 gives necessary and sufficient conditions that an expression containing sentential variables and unary and binary sentential connectives be a formula in the Lukasiewicz notation.This paper extends his result to expressions containing w-ary symbols for all n ^ 0. Sentential variables and constants are treated as the case tt = 0.An expression is a sequence Si • • • Sk such that Si for i= 1, • • • , k is an w-ary symbol for some n.An initial segment of such an expression is an expression $i • • • Si where i 1.A formula is a sequence sz\ • • • z n where s is an w-ary symbol, and Zi, • • • , z n are formulas.The measure [s] of an w-ary symbol s is # -1.The measure [si • • • Sk] of an expression Si • • • Sk is [si] + •••+[**]• THEOREM.Necessary and sufficient conditions that an expression X = Si • • • Sk be a formula are: